According to Mukai [Mu] any smooth curve of genus 8 and Clifford index 3 is the transversal intersection C=ℙ7 ∩ G(2,6) ⊂ ℙ15. In particular this is true for the general curve of genus 8. Picking 8 points in the Grassmannian G(2,6) at random and ℙ7 as their span gives the result.
i1 : FF=ZZ/10007;S=FF[x_0..x_7]; |
i3 : (I,points)=randomCanonicalCurveGenus8with8Points S; |
i4 : betti res I 0 1 2 3 4 5 6 o4 = total: 1 15 35 42 35 15 1 0: 1 . . . . . . 1: . 15 35 21 . . . 2: . . . 21 35 15 . 3: . . . . . . 1 o4 : BettiTally |
i5 : points o5 = {ideal (x + 2218x , x - 4540x , x - 626x , x - 4294x , x + 1276x , 6 7 5 7 4 7 3 7 2 7 ------------------------------------------------------------------------ x - 2638x , x - 2532x ), ideal (x + 3421x , x - 3522x , x - 3180x , 1 7 0 7 6 7 5 7 4 7 ------------------------------------------------------------------------ x + 1692x , x + 1898x , x - 276x , x - 2805x ), ideal (x + 2537x , 3 7 2 7 1 7 0 7 6 7 ------------------------------------------------------------------------ x + 3976x , x - 4098x , x + 4833x , x + 2403x , x + 3542x , x - 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ 2455x ), ideal (x + 4468x , x - 922x , x - 613x , x + 4475x , x - 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ 448x , x - 972x , x + 4770x ), ideal (x - 4035x , x - 1606x , x - 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ 3277x , x + 2598x , x + 3081x , x + 4219x , x - 3350x ), ideal (x + 7 3 7 2 7 1 7 0 7 6 ------------------------------------------------------------------------ 4479x , x - 467x , x + 2935x , x + 885x , x + 1579x , x - 548x , x 7 5 7 4 7 3 7 2 7 1 7 0 ------------------------------------------------------------------------ + 126x ), ideal (x + 4996x , x - 3771x , x - 4847x , x - 4789x , x 7 6 7 5 7 4 7 3 7 2 ------------------------------------------------------------------------ + 648x , x - 4446x , x - 4463x ), ideal (x - 3955x , x + 2510x , x 7 1 7 0 7 6 7 5 7 4 ------------------------------------------------------------------------ - 3643x , x + 2825x , x + 1886x , x - 2116x , x - 2821x )} 7 3 7 2 7 1 7 0 7 o5 : List |